I am a graduate student teaching college algebra at a larger state school, and currently I'm covering transformations of graphs of function, i.e.:

Given the graph of a function $y =f(x)$, what do the following graphs look like, compared to the graph of $y = f(x)?

$y = f(x) \pm C$,

$y = f(x \pm C)$,

$y = Cf(x)$,

$y = f(Cx)$,

$y = -f(x)$,

$y = f(-x)$.

This is a topic that I always struggle to come up with concrete “real world” applications of. My audience is typically freshman who are looking to get the class out of the way, so they can move on to business calculus or so they can be done with math altogether. Consequently, teaching material through a real scenario is a good way to hold my students attention.

However with this topic, I have a hard time coming up with good examples. One such example I gave was the following: I presented them with the graph $y = f(x)$, where $f(x)$ is the function that takes in the number of miles/hour over the speed limit a driver is going and returns the value of the corresponding fine for speeding in a particular city. I then asked them how the graph would change as well as how they could modify $f(x)$ using these elementary transformations if (a) the city decides to crack down on speeding and raise all fines amount by \$50 ( $y = f(x) + 50$) and (b) the city decides to be more lenient and give drivers a 5 mph “buffer” ($y = f(x-5)$) – meaning that a driver would not receive a fine unless they exceeded the speed limit by more than 5 mph.

Does anyone else have better examples that are rooted in a real scenario?

$\begingroup$Not even close to a complete answer, but either pre- or post-composition with a scaling corresponds to a change of units (e.g. feet to inches; kilograms to pounds; etc).$\endgroup$
– Xander HendersonJul 13 '18 at 1:17

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$\begingroup$I think a common example in the spirit of @XanderHenderson is some function $f(t)$ where $t$ has units "year" versus $g(t)=f(t+2000)$ where the $t$ on the left can be thought of as having units "years since 2000".$\endgroup$
– ncrJul 15 '18 at 4:58

2 Answers
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My favorite example of a graphical transformation is waves. Consider,
$$ y = A \sin( x-vt) $$
Here $A$ gives the amplitude of the wave and it tells us from a graphical perspective how much the unit-sine wave is vertically stretched. If we fix $t$ then the term $-vt$ is just some phase-shift and we can see the graph is just a horizontal shift of the sine wave. In particular we shift the $y = A \sin(x)$ graph $vt$-units to right to form $y = A \sin(x-vt)$. This is a right-moving wave if we assume $v>0$. To see it, I think setting $0 \leq x-vt \leq 2\pi$ might be helpful, then $vt \leq x \leq 2\pi+vt$ and if we animate $t$ we can actually see the wave travel.

Of course, adding graphs has interesting interpretations in terms of constructive and destructive interference. Or, if we add time I think you can get beats. There is much to explore here.

Edit: as Dave Renfro pointed out, I am using forbidden trigonometry. To, give a less exciting, but very much applied example consider:
$$ y = y_o- \frac{g}{2}t^2 $$
If we start with $y=t^2$ then multiplying b $-g/2$ stretches the graph and flips it over. The significance is that $g$ is the acceleration due to gravity. Then, adding $y_o$ shifts the graph $y=-gt^2/2$ up by $y_o$ and in this discussion $y_o$ has the significance of being the initial condition.

A more ambitious formula to explain would be $y = y_o+v_ot-\frac{g}{2}t^2$. Completing the square gives,
$$ y = y_o+\frac{v_o^2}{2g}-\frac{g}{2}\left( t-\frac{v_o}{g} \right)^2 $$
We can view this as the graph $y=-\frac{g}{2}t^2$ which is shifted vertically by $y_o+\frac{v_o^2}{2g}$ and translated horizontally by $\frac{v_o}{g}$. If $v_o>0$ then $\frac{v_o}{g}>0$ so it would signify a right-shift of the graph. The vertex of the parabola would be at $\frac{v_o}{g}$ which is where the maximum height of $y_o+\frac{v_o^2}{2g}$ is attained. In contrast, $y=-\frac{g}{2}t^2$ has maximum height $y=0$ at $t=0$. To simplify this discussion, you could say it is all taking place on the Dwarf Moon of Convenience where $g/2=1$.

$\begingroup$Ordinarily this would be an excellent example, but unfortunately Oiler is teaching students who will be taking business calculus, which almost always has no trace of trigonometry. Personally, I found the greatest help in beginning calculus classes is simply being able to make quick rough sketches of graphs of functions on the fly, such as $y = 2 - x^2$ and $y = x^2 - 1$ when you want to find the area bounded by these two graphs.$\endgroup$
– Dave L RenfroJul 13 '18 at 15:32

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$\begingroup$Indeed, the "applied calculus" or "business calculus" has all the fun stripped out of it. I guess he could still use a piece-wise-defined function to make a triangle or square wave. Actually, you gave me an idea, I'll edit.$\endgroup$
– James S. CookJul 13 '18 at 15:44

Unit conversions often provide a natural application of these transformations. Since temperature unit conversions often include both scaling and shifting, they are particularly useful.

For example:

Suppose you have a function $H(c)$ which gives the rate of heat stroke death in a region, per thousand occupants, as a function of the temperature $c$, in Celsius. You wish to change units to have the input of the temperature be in Fahrenheit, represented by $f$, and to instead output the rate of heat stroke death per hundred thousand people. Your new function would be $100 H(\frac{5}{9}(f-32))$. What would the graph of the new function look like?

Suppose you have a function $C(d)$ giving the temperature $C$, in Celsius, on day $d$ of the year. You wish to instead input the approximate month, m, of the year, considering each month to be 30 days, and having m=0 represent the first day of January. You wish to have your function output the temperature in degrees Fahrenheit below 90 degrees (since, perhaps, this value is what determines the heating costs of your 90-degree greenhouse). Your new function will be of the form $ 90- (\frac{9}{5}C(30m)+32)$. What will the graph of your new function look like?

Of course, these are intentionally challenging examples that are intended to pack as much possible into one problem. Some other specific ways to incorporate each individual transformation could include:

$f(x)\pm C$: Surcharges and flat fees. Measuring altitude vs. height above a fixed altitude, e.g. "The tree line is ____, and located 10,000 feet in altitude. If $A(t)$ is the altitude of a climber at time t, write a function describing their height above the tree line at time t."

$f(-x)$: Instead of measuring time "after" an event measure time "before" an event.

$f(Cx)$: Convert the units of the input of the function by a scalar, e.g. feet to inches

$\begingroup$I was just about to post something similar, I was thinking for business currency or asset conversion might work? Something like the following: Gold can be bought at a rate of $g(d) = .01 d^2$ ounces per $d$ USD (up to a cap of 1000 oz). If the current exchange rate is $.75$ CAD to 1 USD, what function $f(d')$ gives the number of pounds of gold that can be bought for $d'$ CAD? How much does one ounce of gold cost in Canadian dollars? How does that compare to the price of an ounce of gold in US?$\endgroup$
– Nate BadeJul 13 '18 at 19:29

$\begingroup$In the first example I think the scaling factor should be $\frac{1}{100}$ rather than $100$, unless by "rate of heat stroke death, per thousand occupants" you mean something like "deaths per unit time, per thousand occupants".$\endgroup$
– Peter TaylorJul 16 '18 at 12:49